Abstract
Let be a diffeomorphism on a n-dimensional manifold. Let be the chain component of f associated to a hyperbolic periodic point p. In this paper, we show that (i) if f has the -stably orbitally shadowing property on the chain recurrent set , then f satisfies both Axiom A and no-cycle condition, and (ii) if f has the -stably orbitally shadowing property on , then is hyperbolic.
MSC:37C50, 34D10, 37C20, 37C29.
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1 Introduction
Let M be a closed n-dimensional manifold, and let be the space of diffeomorphisms of M endowed with the -topology. Denote by d the distance on M induced from a Riemannian metric on the tangent bundle TM. Let . For , a sequence of points () in M is called a δ-pseudo-orbit of f if for all . Let be a closed f-invariant set. We say that f has the shadowing property on Λ (or Λ is orbitally shadowable) if for every , there is such that for any δ-pseudo-orbit of f (), there is a point such that for all . It is easy to see that f has the shadowing property on Λ if and only if has the shadowing property on Λ for . The notion of pseudo-orbits often appears in several methods of the modern theory of a dynamical system. Moreover, the shadowing property plays an important role in the investigation of stability theory and ergodic theory. Actually, in [1, 2], the authors showed that every f satisfying both Axiom A and the strong transversality condition has the shadowing property. Since such a system is structurally stable, there is a -neighborhood of f such that every has the shadowing property because f satisfies both Axiom A and the strong transversality condition. And in [3], Sakai proved that if there is a -neighborhood of f, for any , g has the shadowing property, then f satisfies both Axiom A and the strong transversality condition.
For each , let be the orbit of f through x; that is, . We say that f has the orbital shadowing property on Λ (or Λ is orbitally shadowable) if for any , there exists such that for any δ-pseudo-orbit , we can find a point such that
where denotes the ϵ-neighborhood of a set . f is said to have the weak shadowing property on Λ (or Λ is weakly shadowable) if for any , there exists such that for any δ-pseudo-orbit , there is a point such that .
The orbital shadowing property is a weak version of the shadowing property: the difference is that we do not require a point of a pseudo-orbit ξ and the point of an exact orbit to be close ‘at any time moment’; instead, the sets of the points of ξ and are required to be close. The weak showing property is a slightly weak version of the orbital shadowing property. The difference is that a set of points of a ‘sufficiently precise’ pseudo-orbit ξ is required to be contained in a small neighborhood of some exact orbit . We say that Λ is locally maximal if there is a compact neighborhood U of Λ such that
It is easy to see that f has the orbital shadowing property on Λ if and only if has the orbital shadowing property on Λ for .
Now we introduce the notion of the -stably orbitally shadowing property on Λ. We say that f has the -stably orbitally shadowing property on Λ if there are a -neighborhood of f and a compact neighborhood U of Λ such that (i) (locally maximal), (ii) for any , g has the orbital shadowing property on , where is the continuation of Λ. We say that f has the -stably orbitally shadowing property if in the above definition. It is known that if any structurally stable diffeomorphism f has a -neighborhood of f, then for any , g has the shadowing property, hence g has the orbital shadowing property. In [4], the authors showed that if f has the -stably orbitally shadowing property, then f satisfies both Axiom A and the strong transversality condition. Thus we can restate the above facts as follows.
Theorem 1.1 [4]
Let be a diffeomorphism. f has the -stably orbitally shadowing property if and only if f is structurally stable.
For any , we write if for any , there is a δ-pseudo orbit () of f such that and . The set of points is called the chain recurrent set of f and is denoted by . It is well known that is a closed and f-invariant set. If we denote the set of periodic points of f by , then . Here is the non-wandering set of f. We write if and . The relation ↭ induces on an equivalence relation, whose classes are called chain components of f. Note that by [5], the map is upper semi-continuous. From the fact, we will show the following result.
Theorem 1.2 Let be the chain recurrent set of f. f has the -stably orbitally shadowing property on if and only if f satisfies Axiom A and the no-cycle condition.
Let f be an Axiom A diffeomorphism. Then, it is well known that if and only if f satisfies the no-cycle condition. Hence, f has the -stably orbitally shadowing property on and is characterized as the Ω-stability of the system by Theorem 1.2.
We say that Λ is hyperbolic for f if the tangent bundle has a Df-invariant splitting and there exist constants and such that
for all and . If , then f is Anosov.
If is a hyperbolic saddle with the period , then there is no eigenvalues of with modulus equal to 1, at least one of them is greater than 1, at least one of them is smaller than 1. Note that there is a -neighborhood and a neighborhood U of p such that for all , there is a unique hyperbolic periodic point of g with the same period as p and . Here , and the point is called the continuation of p. The stable manifold and the unstable manifold are defined as follows. It is well known that if p is a hyperbolic periodic point of f with period k, then the sets
are -injectively immersed submanifolds of M.
Denote by the chain component of f containing p. If p is a sink or source periodic point, then is a periodic orbit itself. Therefore, in this paper, we may assume that all periodic points are the saddle type. Let . We say that p and q are homoclinic related, and write if , and . It is clear that if then .
Denote by the homoclinic points associated with p, that is, , and let be the transverse homoclinic points associated with p, that is, . Obviously, and are closed f-invariant sets, and it is clear that . Note that by Smale’s transverse homoclinic point theorem, coincides with the closure of the set of all such that . It is known that is a transitive set, and if is not hyperbolic, then it may contain periodic points having different indices. Note that (see [[6], examples and counter examples]). For the chain component , Lee, Moriyasu and Sakai [7] showed that f has the -stably shadowing property on if and only if the chain component is a hyperbolic homoclinic class of p. From the fact, we consider the orbital shadowing property and the chain component. The following result is the main theorem in this paper.
Theorem 1.3 Let p be a hyperbolic periodic point of f, and let be the chain component of f containing p. f has the -stably orbitally shadowing property on if and only if is the hyperbolic homoclinic class of p.
If is hyperbolic, then it is locally maximal, that is, there is a compact neighborhood U of such that . By the local stability of a hyperbolic set, there is a -neighborhood of f such that for any , is hyperbolic. Then we know that g has the shadowing property on , and so g has the orbital shadowing property on . Thus we get the ‘if’ part. Thus, we need to show that f has the -stably orbitally shadowing property on , then is hyperbolic.
2 Proof of Theorem 1.2
Let M be as before, and for , we denote by the closed ϵ-ball of a subset A of M.
Proposition 2.1 Let . Suppose that f has the -stably orbitally shadowing property on . Then f satisfies Axiom A and the no-cycle condition.
Denote by the set of homeomorphisms of M. For the proof of the following lemma, see [5].
Lemma 2.2 Let , and let be the chain recurrent set of f. For any , there is such that if (), then .
The following Franks’ lemma will play essential roles in our proofs.
Lemma 2.3 [8]
Let be any given -neighborhood of f. Then there exist and a -neighborhood of f such that for given , a finite set , a neighborhood U of and linear maps satisfying for all , there exists such that if and for all .
Denote by the set of such that there is a -neighborhood of f with the property that every () is hyperbolic. It is proved by Hayashi [9] that if and only if f satisfies both Axiom A and the no-cycle condition. Let be an invariant submanifold of f. We say that Λ is normally hyperbolic if there is a splitting such that
-
(a)
the splitting depends continuously on ,
-
(b)
() for all ,
-
(c)
there are constants and such that for every triple of unit vectors , and (), we have
for all .
Proof of Proposition 2.1 First we suppose that f satisfies both Axiom A and the no-cycle condition. Then is hyperbolic, and so is locally maximal. By the stability of locally maximal hyperbolic sets, we can choose a compact neighborhood U of and a -neighborhood of f such that , and for any , is hyperbolic for g. Since is hyperbolic for f, f has the shadowing property on , and so f has the orbital shadowing property on . Thus, we know that f has the -stably orbitally shadowing property on . Finally, we show that if f has the -stably orbitally shadowing property on , then f satisfies both Axiom A and the no-cycle condition. From the above facts, to complete the proof of the theorem, it is enough to show that if f has the -stably orbitally shadowing property on , then .
Suppose that f has the -stably orbitally shadowing property on . Then there are a compact neighborhood U of and a -neighborhood of f such that for any , g has the orbital shadowing property on , where is the continuation of Λ. Choose satisfying . By Lemma 2.2, there is such that if for , then
where is the usual -metric on . Put . Then for each , and so for all . This means that for . Since g has the orbital shadowing property on , g has the orbital shadowing property on .
Let , and let be a -neighborhood of f which is given by Lemma 2.3 with respect to . Let , and let k be the period of p. Let
where , , are -invariant subspaces corresponding to eigenvalues λ of for , and , respectively. It is sufficient to show that each is hyperbolic. Suppose this is not true. Then there exist eigenvalues of with or , . Let be the -ball of f. Set . For , we can obtain a linear automorphism such that
(a.i) ,
(a.ii) keeps invariant, where ,
(a.iii) all eigenvalues of , say , , are roots of unity.
Let F be a finite set . Define
Observe that . Thus for all . By Lemma 2.3, we can find a diffeomorphism and such that
(b.i) ,
(b.ii) , , ,
(b.iii) on , and
(b.iv) on , .
Define
Then by (a.iii) we can find such that . Choose a small satisfying such that
where . Then by (b.iv) we have
on . Since , we get
We write
where , . Then is -invariant. If the eigenvalue is real, then and is an arc centered at p; and if the eigenvalue is complex, then and is a disk centered at p. We know that and . By (1), we get
and
Since g has the orbital shadowing property on , g must have the orbital shadowing property on and . Since and are normally hyperbolic, we can see that a shadowing point is in and . Observe that on . By our construction, is the identity on the arc as well as on the disk . Since the identity map does not have the orbital shadowing property. Indeed, let be a δ-pseudo orbit. Then, by the orbital shadowing property and normally hyperbolicity, there is a point such that and . But since g is the identity map on for all , . Thus we have . This is a contradiction. This completes the proof of Proposition 2.1. □
3 Proof of Theorem 1.3
Let , and let be a hyperbolic saddle with period . Then there are the local stable manifold and the unstable manifold of p for some . It is easily seen that if , for all , then , and if for all , then . Note that the local stable manifold (resp. the local unstable manifold ).
Lemma 3.1 Let p be a hyperbolic periodic point of f, and let be the chain component of f containing p. If f has the orbital shadowing property on , then .
Proof We know that . We now show that . Let . Then and . Thus, for any , there is a periodic η-pseudo orbit of f such that , and for some and . By [[10], Proposition 1.6], . To simplify, we may assume that . Then we extend the pseudo orbit as follows:
-
(i)
for all and
-
(ii)
for all .
Thus we see that as . Then we get an η-pseudo orbit:
Since p is a hyperbolic periodic point of f, we can take an such that for all implies and for all implies . Take , and let be the number in the definition of the orbital shadowing property for f. From the above, we set . Then we see that . Since f has the orbital shadowing property on , there is such that
There are and such that , and .
Then for all , and for all . Hence and . Therefore, we see that . Thus we can find such that . □
Let Λ be a closed f-invariant set. Note that if f has the orbital shadowing property on a locally maximal Λ, then the shadowing point can be taken from Λ.
Lemma 3.2 Suppose that f has the -stably orbitally shadowing property on Λ. Then for every is hyperbolic.
Proof Suppose that f has the -stably orbitally shadowing property on Λ. Then there exist a compact neighborhood U of Λ and a -neighborhood of f such that for any , g has the orbital shadowing property on . Take a -neighborhood of f as in Lemma 2.3. Then arguing similarly as in the proof of Theorem 1.2, is seen to be a desired neighborhood of f. In fact, suppose is a non-hyperbolic periodic point for some . Observe that we can choose smaller if necessary so that . Then, by making use of Lemma 2.3, we can construct a diffeomorphism -close to g which has the invariant hyperbolic small arc and disk , centered at q, contained in , where . Note that for some , where either or . Since the identity map does not have the orbital shadowing property and has the orbital shadowing property on as well as on , we have a contradiction. This completes the proof. □
For , we say that a compact f-invariant set Λ admits a dominated splitting if the tangent bundle has a continuous Df-invariant splitting and there exist , such that for all and , we have
Remark 3.3 If Λ admits a dominated splitting such that for any , is constant, then there are a -neighborhood of f and a compact neighborhood V of Λ such that for any , admits a dominated splitting with .
From Lemma 3.2, for any , the family of periodic sequences of linear isomorphisms of generated by Dg along the hyperbolic periodic points is uniformly hyperbolic (see [11]). Indeed, there is such that for any , , and any sequence of linear maps with for , and is hyperbolic. Here is the -neighborhood of f. Then we can apply Proposition 2.1 in [11] to obtain the following proposition.
Proposition 3.4 Suppose that f has the -stably orbitally shadowing property on . Then there exist a -neighborhood of f, constants , and such that
-
(1)
for each , if q is a periodic point of g in with period (), then
where .
-
(2)
admits a dominated splitting with .
Remark 3.5 By Proposition 3.4(2) and [[12], Theorem A], the homoclinic class is the transverse homoclinic class . Thus if f has the -stably orbital shadowing property on , then we see that .
Let j denote the , , and let
where . We write . Then we know that these are basic sets and . In general, a non-hyperbolic homoclinic class contains saddle periodic points with different indices. Since , we know that the chain component may contain saddle periodic points with different indices in general. However, if f has the -stably orbitally shadowing property on , then such a case cannot happen. Thus we need the following proposition to prove Theorem 1.3.
Proposition 3.6 Suppose that f has the -stably orbitally shadowing property on . Then for any , .
To prove Proposition 3.6, we need the following lemma.
Lemma 3.7 Suppose that f has the orbital shadowing property on . Then, for any hyperbolic ,
Proof Suppose that f has the orbital shadowing property on . Let and be as in the definition of and with respect to p and q. To simplify notation in this proof, we may assume that and . Take , and let be the number of the definition of the orbital shadowing property for f.
For , there is a finite δ-pseudo orbit () such that
-
(i)
, and
-
(ii)
, and .
We extend the finite δ-pseudo orbit as follows: Put
-
(i)
for all , and
-
(ii)
for all .
Then we get a δ-pseudo orbit
Since f has the orbital shadowing property on , there is a point such that
Then, we see that . Thus . The other case is similar. □
A diffeomorphism f is said to be Kupka-Smale if the periodic points of f are hyperbolic, and if , then is transversal to . It is well known that the set of Kupka-Smale diffeomorphisms is a -residual set in .
Proof of Proposition 3.6 Suppose that f has the -stably orbitally shadowing property on . Let be a -neighborhood of f and U be a compact neighborhood of as in the definition. Note that is upper semi continuous and is lower semi continuous. By Lemma 3.1, , is semi continuous. By the definition, . To derive a contradiction, we may assume that there is a point such that for any , and , where is the continuation. By Lemma 3.2, for every is hyperbolic. Then we know that . Take a Kupka-Smale diffeomorphism . Then there are the and that are the continuations of p and q respectively, and . Since and , we know that , where and are the stable and the unstable manifolds of and with respect to g. On the other hand, since , g has the orbital shadowing property on . Thus g has the shadowing property on . By Lemma 3.7, . This is a contradiction. □
Let us recall Mañé’s ergodic closing lemma obtained in [11]. Denote by an ϵ-tubular neighborhood of ; that is,
Let be the set of points such that for any -neighborhood and , there are and such that on and , for . The following lemma is in [11].
Lemma 3.8 [11]
For any f-invariant probability measure μ, we have .
End of the Proof of Theorem 1.3 Suppose that f has the -stably orbitally shadowing property on . Let be the -neighborhood of f given by Proposition 3.4. To get the conclusion, it is sufficient to show that is hyperbolic, where , and i is the index of p. Fix any neighborhood of . Note that by Proposition 3.6, if .
Thus we show the following: Let be a small connected -neighborhood of f. If any satisfies on , then for any . Indeed, suppose not, then there are and such that on and . Suppose that , , and define by
By Lemma 3.2, the function γ is continuous, and since is connected, it is constant. But the property of implies . This is a contradiction.
Finally, to prove Theorem 1.3, we use the proof of Theorem B in [11]. Thus we show that
for all , and thus, the splitting is hyperbolic.
More precisely, we will prove the case of (the other case is similar). It is enough to show that for any , there exists such that
We will derive a contraction. If it is not true, then there is such that
for all . Thus
for all . Define a probability measure
Then there exists () such that , as , where M is a compact metric space. Thus
By Mañè [[11], p.521],
where is an f-invariant measure. Let
and = {, there exist and such that on and for }.
Note that if , such that as , then . So, this is a contradiction.
For any , . Then, for any ,
since and . Thus, almost everywhere. Therefore,
By Birkhoff’s theorem and the ergodic closing lemma, we can take such that
By Proposition 3.4, this is a contradiction.
Thus, by Proposition 3.4, .
Let , and be given by Proposition 3.4 and take and such that
Then, by Mañé’s ergodic closing lemma, we can find on and near by r.
Moreover, we know that since on . By applying Lemma 2.3, we can construct () -nearby g such that
(see [[11], pp.523-524]). On the other hand, by Proposition 3.4, we see that
We can choose the period () of as large as . Here . This is a contradiction. Thus,
for all . Therefore, is hyperbolic. This completes the proof of the ‘only if’ part of Theorem 1.3. □
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Acknowledgements
The author wishes to express his deep appreciation to the referee for his careful reading of the manuscript. This work was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (No. 2011-0007649).
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Lee, M. Chain components with -stably orbital shadowing. Adv Differ Equ 2013, 67 (2013). https://doi.org/10.1186/1687-1847-2013-67
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DOI: https://doi.org/10.1186/1687-1847-2013-67